Parana Journal of Science and Education publication info —A NEW PHYSICAL CONSTANT FROM THE RATIO OF THE RECIPROCAL OF THE "RYDBERG CONSTANT" TO THE PLANCK LENGTH— Ozan Yarman1, Ricardo Gobato2, Tolga Yarman3, Metin Arık4 Manuscript prepared via LATEX April 2018 Abstract. This study presents a unique set of solutions, using empirically determined physical quantities, in achieving a novel dimensionless constant α(1/R 1)/PL from the ratio of the inverse of the Rydberg constant to the Planck length. It is henceforth shown that the Lorentz Scalar coming into play, which we dub the Parana constant, necessitates us to interpret the Gravitational constant G as being neither universal nor Lorentz Invariant. Just the same, the elementary charge in the MKS system should not by 2 itself be considered as Lorentz Invariant, but the term e /"0, including its powers, ought to be. That being the case, the "Rydberg constant" must not, according to the present undertaking, be deemed a ubiquitous magnitude either, but the ratio of its reciprocal to Planck length would, in effect, be. The Parana constant is furthermore shown to exhibit meaningfulness as the proportion of the Planck mass to the electron rest mass. Throughout our derivations, we take the oppurtunity to reveal interesting features and deliberate over them. Keywords: Rydberg constant, Planck length, Lorentz Invariance, Lorentz Scalar, Dimensionless Physical Constant, Parana constant. AMS Subject Classification: 81V45 1 Prof. Dr., İstanbul University, İstanbul, Turkey. e-mail: [email protected]; 2 Prof. Dr., Laboratory of Biophysics and Molecular Modeling Genesis, Parana, Brazil. 3 Prof. Dr., Okan University, İstanbul Turkey. 4 Prof. Dr., Boğaziçi University, İstanbul Turkey. § Manuscript ©: 15 April 2018. Parana Journal of Science and Education. 2 1. Introduction The Universal Matter Architecture (UMA) scaffolding [1, 2, 3, 4, 5, 6, 7] developed a few decades ago by the third co-author, which led him to derive Yarman’s Approach [8, 9] for all force interactions that resulted in YARK (Yarman-Arık-Kholmetskii) gravitation theory [10, 11, 12, 13, 14, 15, 16] (where a symbiosis between Quantum Mechanics and gravitation was harmoniously achieved — with the associated gravitational field energy becoming a non-vanishing quantity in all possibly definable reference frames), makes certain that the "theoretical speed of light in vacuum" c0 (exactly 299,792,458 m/s) is "truly" a universal constant just like the Planck constant h (having the dimensions Js or m2 kg s−1) and the elementary charge in the CGS unit p system (bearing the dimensions statCoulomb = cm3=2 g s−1). In other words, these quantities remain rigorously invariant under Lorentz Transformations when embedded within even a strong gravitational field of an immensely massive stationary body, or any other type of field an object under consideration would interact with. However, as it shall soon be disclosed, "the elementary charge in the MKS system" — with this simply meaning a specific Ampere times 1 second, where the MKS unit of electrical current "Ampere" is determined solely via the arbitrary assignment of the −7 2 value of the magnetic permeability of classical vacuum in history as µ0 = 4π ∗ 10 N/A owing to the choice of placing two parallel electrical wires of ideal length and of negligible cross-section a meter apart to achieve 2 ∗ 10−7 Newtons attraction force between them (so long as a homogenous flow of charged particles in each is maintained) — should not by 2 itself be thought of as Lorentz Invariant or constant, but the term e /"0, as well as its exponents, ought to be (where "0 denotes the permittivity of free space)[17]. In a similar vein, the present undertaking necessitates an interpretation of the Gravitational constant G (possessing the dimensions m3 kg−1s−2; and otherwise understandable as "acceleration of surface area per unit mass" from an extrapolation of its dimensions [L2/M] * [L/T2]) as being neither universal nor Lorentz Invariant [cf. 18, 19, 20, 21]. That being the case, we may refer to it as GC in order to delineate a flexible Earth-bound magnitude. Finally, with the elementary charge value in the MKS system and the "Gravitational constant" failing to represent universal quantities, followed by the implication that the electron rest mass (me0) should too get altered commensurately with force intensity as per Yarman’s Approach [22, 23], we find the "Rydberg constant" to be not at all a ubiquitous invariant either. The principal way to demonstrate these facts will be through the set of derivations below of a novel physical constant α(1/R 1)/PL as the ratio of the reciprocal of the "Rydberg constant" to the Planck length [24]. It will be seen that this new Lorentz Scalar has profound meaning for natural philosophy. 3 2. Inverse of the Rydberg constant to the Planck length: A new dimensionless invariant quantity To begin with, let us equate, by a factor of 1/n, the reciprocal of the known Rydberg constant (making thus 1/10,973,731.5705365 meters) to the known Planck length in the MKS unit system (considered to be 1.61639446559731E-35 meters) — all the while remembering that "0 is classically (that is to say, following the era of J. C. Maxwell [25]) 2 1 derived from the magnetic permeability of vacuum µ0 owing to the relationship c = µ0"0 7 as 10 Farads per meter, where |c|2 is the modulus of the square of the speed of light 4πjcj2 in empty space (just the number 299,792,4582). Therefore: 8 ∗ h3 ∗ c ∗ " 2 r 0 0 = Gh ; (1a) 4 3 n ∗ me ∗ e 2πc0 82 ∗ h6 ∗ c2 ∗ 10(7∗4) F4 Gh 2 2 8 2 4 4 = 3 ; (1b) n ∗ me ∗ e ∗ (4πjcj ) meter 2πc (2 ∗ 2 ∗ 2)2 ∗ h¡65 ∗ c5 ∗ 1028 ∗ ¨2¨π 4 F = n 2; (1c) 2 8 ¨ 2 2 3 4 me ∗ e ∗ (2 ∗ ¨2πjcj ) ∗ (4πjcj ) ∗ G ∗ h meter 5 5 28 4 −4 (4¡ ∗ 4¡ ∗ 4)¡ ∗ h ∗ c ∗ 10 F ∗ meter 2 2 8 2 2 2 2 = n ; (1d) G ∗ me ∗ e ∗ (2 ∗ jcj ) ∗ (4¡πjcj ) ∗ (4¡πjcj ) ∗ (4¡πjcj ) 5 5 28 4 −4 h ∗ c ∗ 10 F ∗ meter 2 2 8 3 8 = n ; (1e) G ∗ me ∗ e ∗ 2 ∗ π ∗ jcj h5 ∗ c 5 ∗ 1028 4 ∗ meter−4 ∗ meter5 F = n 2; (1f) 2 8 3 5 3 5 G ∗ me ∗ e ∗ 2 ∗ π ∗ jcj ∗ jcj ∗ second 5 4 −4 28 5 2 h ∗ F ∗ meter 10 ∗ meter n = 2 8 ∗ 3 3 5 ; (1g) G ∗ me ∗ e 2 ∗ π ∗ jcj ∗ second 4 5 5 4 2 h ∗ v ∗ 1 F n = 2 8 4 ; (1h) G ∗ me ∗ e ∗ 1 meter r 2 h5 v5 1 F n = ∗ 4 2 ; (1i) GC me ∗ e ∗ 1 meter r 28 5 where v= 5 10 ∗ meter = 1:43024900891066 m/s (indicating here an 2 ∗ π3 ∗ 2997924583 ∗ second5 "enigmatic velocity"), with n corresponding to our novel dimensionless quantity α(1/R 1)/PL=5.63765262613852E+27 (i.e., Parana constant) and 1/n thus making α PL/(1/R 1)=1.77378789775656E-28 if we rely on the latest values (without highlighting the related measurement uncertainties) me =9.1093835611E-31 kg; (2a) h =6.62607004081E-34 Js; (2b) c0 = 299; 792; 458 (m=s); (2c) e =1.602176620898E-19 C; (2d) "0 =8.854187817620E-12 (F=m); (2e) 3 −1 −2 GC =6.6754518E-11 (m kg s )[19]: (2f) Upon the revelation we have landed on at this stage (with GC representing an Earth-bound quantity), it is right away possible to crosscheck the one-to-one dimensional correspondence of the LHS to the RHS of the relationship emerging from Eq. (1h) 8 m3 ∗ kg4 ∗ 4 C ≡ F ; (3a) v5 s3 5 ergo, m8 ∗ kg4 ∗ 4 8 ≡ F ; (3b) C s8 r 8 m8 ∗ kg4 ∗ 4 ≡ F ; (3c) C s8 r 8 m12 ∗ kg4 ∗ 4 ≡ F ; (3d) C s8 ∗ meter4 p m(3=2) kg pC ≡ : (3e) " s The last proportionality is none other than the MKS equivalent of the Lorentz Invariant CGS unit StatCoulomb (otherwise christened the "electrostatic unit" orp ESU in the literature). The exact transformation to ESU (bearing the dimensions of ~c) is achieved via: p p 107+2 ∗ m(3=2) kg p C0 = 10 ∗ jc j ∗ ; (4a) −2 −2 0 100 ∗ 1000 ∗ 4π"0 s p p 109 e cm(3=2) g p MKS = 4:80320467329146E-10 (StatC): (4b) 4π"0 s Hence, we have straightforwardly ascertained that it is not 1:602176620898 · 10−19 C that is a universal constant, but instead p p ( 109 ∗ 1:602176620898 · 10−19 C)/( 4π ∗ 8:854187817620 · 10−12 F=m) that equates to Newtons force times surface area in squaremeters (i.e., Nm2, which otherwise bears the dimensions of hc the way these appear in the classical expression of the Fine Structure 2 p constant α = e ); with the factor 10 in 107+2 to counterbalance the 10 on the RHS 2"0hc of Eq. (4a) obviously not representing anything physical in the conversion. Proceeding from Eq. (1g), we are able to find two alternative equalities that yield the same dimensionless value for α(1/R 1)/PL: 5 4 −4 28 5 2 h ∗ F ∗ meter 10 ∗ meter n = 2 8 ∗ 3 3 5 ; (5a) G ∗ me ∗ e 2 ∗ π ∗ jcj ∗ second 6 5 (7∗4) 4 5 2 h ∗ 10 ∗ F ∗ meter n = 2 8 3 3 5 4 ; (5b) G ∗ me ∗ e ∗ 2 ∗ π ∗ jcj ∗ second ∗ meter 107 ∗ 10(7∗3) ∗ h5 ∗ jcj3 ∗ 4 ∗ meter¡54 n 2 = F ; (5c) 2 8 3 6 5 43 G ∗ me ∗ e ∗ 2 ∗ π ∗ jcj ∗ second ∗ meter¡ 3 7 3 5 4 (7∗3) 3 2 4 ∗ 10 ∗ jcj ∗ h ∗ F ∗ meter 10 ∗ F n = 2 8 5 ∗ 2 3 3 ; (5d) G ∗ me ∗ e ∗ 2 ∗ second (4πjcj ) ∗ meter 32 ∗ 107 ∗ c 3 ∗ h5 ∗ " 3 ∗ ∗ meter¡4 n 2 = 0 0 F ; (5e) 2 8 52 G ∗ me ∗ e ∗ second ¡ s 32 ∗ 107 ∗ meter ∗ " 3 ∗ c 3 ∗ h5 α = 0 0 ; (5f) (1/R 1)/PL 2 8 G ∗ me ∗ e ∗ second ∗ Ω s 7 3 3 5 32 ∗ 10 ∗ meter ∗ "0 ∗ c0 ∗ h n = 2 8 ; (5g) G ∗ me ∗ e ∗ H s 32 ∗ 4π ∗ " 3 ∗ c 3 ∗ h5 n = 0 0 ; (5h) 2 8 −7 H G ∗ me ∗ e ∗ (4π ∗ 10 meter) s 3 5 3 128π ∗ c0 ∗ h ∗ "0 α(1/R 1)/PL = 2 8 ; (5i) G ∗ me ∗ e ∗ µ0 and accordingly, v 8 u2π ∗ c 3 ∗ h5 ∗ " 2 u 0 0 n = ∗ µ ; (5j) Z0 t 2 8 ¡0 G ∗ me ∗ e ∗ ¡ ¡"0 7 3 r 3 8 h "0 c0 α(1/R 1)/PL = 4 ∗ ; (5k) Z0 me e G~ where we once more return to the beginning of this section, since the impedance of free space is also Z0 = 1=("0c0) — thus providing us with the reciprocal of the Rydberg constant on the LHS and the inverse of the Planck length on the RHS of Eq.